An Improved Algebraic Multigrid Method for Solving Maxwell's Equations

Abstract

We propose two improvements to the Reitzinger and Schoberl algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwell's equations. The main focus in the Reitzinger/Schoberl method is to maintain null space properties of the weak $\nabla \times \nabla \times$ operator on coarse grids. While these null space properties are critical, they are not enough to guarantee h-independent convergence of the overall multigrid method. We illustrate how the Reitzinger/Schoberl AMG method loses h-independence due to the somewhat limited approximation property of the grid transfer operators. We present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates. The first improvement is based on smoothing the Reitzinger/Schoberl grid transfer operators. The second improvement is obtained by using higher order nodal interpolation to derive the corresponding AMG interpolation operators. While not completely h-independent, the resulting AMG/CG method demonstrates improved convergence behavior while maintaining low operator complexity.